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Chapter 1

Efficient and Validated Time Domain NumericalModeling of Semiconductor Optical Amplifiers (SOAs)and SOA-based Circuits

Christos Vagionas, Jan Bos, George T. Kanellos, Nikos Pleros andAmalia Miliou

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/61801

Abstract

Semiconductor optical amplifiers (SOAs) have been extensively used in a wealth of tele‐com and datacom applications as a powerful building block that features large opticalgain, all-optical gating function, fast response, and ease of integration with other func‐tional semiconductor devices. As fabrication technologies are steadily maturing towardenhanced yield, SOAs are foreseen to play a pivotal role in complex photonics integratedcircuits (PICs) of the near future. From a design standpoint, accurate numerical modelingof SOA devices is required toward optimizing PICs response from a system perspective,while enhanced circuit complexity calls for efficient solvers. In this book chapter, wepresent established experimentally validated SOA numerical modeling techniques and again parameterization procedure applicable to a wide range of SOA devices. Moreover,we describe multigrid concepts and implicit schemes that have been only recently pre‐sented to SOA modeling, enabling adaptive time stepping at the SOA output, with densesampling at transient phenomena during the gain recovery and scarce sampling duringthe steady-state response. Overall, a holistic simulation methodology approach alongwith recent research trends are described, aiming to form the basis of further develop‐ments in SOA modeling.

Keywords: Semiconductor optical amplifier, numerical modeling, transfer matrix meth‐od, multigrid techniques

1. Introduction

The explosive growth of information traffic and the concomitant increase of bandwidth hungryapplications have contributed to a growing demand for communications networks offering

© 2015 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative CommonsAttribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,and reproduction in any medium, provided the original work is properly cited.

greater bandwidth and flexibility at lower cost [1–3]. This has led to a series of technologicaladvances in high-speed backbone networks and telecommunications, where integrated opticalsystems delivering low-cost, low-power, high-bandwidth transmission, and high-speedswitching elements are increasingly required [4]. Toward meaningful and functional subsys‐tems, high-speed all-optical elements such as all-optical signal wavelength conversion [5], on/off keying modulation [6,7], header recognition [8], optical buffering [9], and signal regener‐ation [10] have been developed, targeting to perform basic functionalities in the opticaldomain. Meanwhile, photonic integration technologies have steadily matured over the lastdecade, achieving improved yield fabrication [11,12], allowing the fabrication of complexsingle-chip photonic modules with advanced functionality. This has resulted in impressivedemonstrations of functional complex circuits and switching architectures [12–17], makingfirm steps toward medium-scale (MS) photonic integrated chips (PICs).

On the way to develop complex devices with enhanced processing capabilities, semiconductoroptical amplifiers (SOAs) have been a powerful building block that provides high-speed all-optical switching operations [18] by featuring large optical gain, optical gating function,controllable performances by injection current, compactness, fast response, and ease ofintegration with other functional semiconductor devices [18]. SOAs are utilized in telecom[19,20] and datacom [21,22] PICs either as single SOA travelling waveguides, supporting cross-gain modulation (XGM) phenomena, or arranged in SOA–Mach–Zehnder interferometer(SOA-MZI) configurations, supporting cross-phase modulation (XPM) phenomena. Thereason lies in the maturity of the SOA technology to a point where commercial devices arereadily available either as bulk single chip elements [23–25], in arrays of certain pitch [26,27],or packaged in discrete fiber pigtailed components [28] for use in optical communicationsystems. Moreover, with recent research achievements on mid-board flip-chip bonding of anSOA array to a silicon-on-insulator (SOI) platform [27] or the development of temperaturestable SOA [25], SOAs are expected to play a pivotal role in future PICs.

In this regime, SOA numerical modeling has attracted a lot of research attention during thelast decades, as mathematical models are required to aid in the design of SOAs and to predicttheir operational characteristics [29–36]. During the development process and prior tofabrication, SOA models that take into account the semiconductor properties of the activematerial and cross-sectional dimensions are employed when optimizing the amplification gainand output power over a wideband steady state and under various external current injections.In addition, during the system-level performance study of SOA-based circuits, time domainsimulations are required for an accurate evaluation of the circuit response with multiple signalspropagating bidirectionally along the waveguide. With growing complexity of circuits, systemlevel designs call for experimentally validated yet efficient solvers that support multiplesignals at different wavelengths propagating in multiple directions.

This book chapter aims to provide a holistic simulation methodology approach for efficientand validated numerical modeling of SOAs and SOA-based circuits in the time, which mayform the basis of further developments in SOA modeling. This chapter is organized as follows:a short review of established SOA modeling approaches in the literature will be presented inSection 2, along with an overview of some recent research efforts. In Section 3, a description

Some Advanced Functionalities of Optical Amplifiers2

of an experimentally validated time domain SOA model relying on the TMM will be presented[37–41], relying on explicit modeling techniques, followed by a gain parameterization proce‐dure for tailoring the TMM model to characterization measurements of other SOA devices inSection 4. Having developed an experimentally validated SOA model, we numericallyinvestigate the impact of the external current injection, the SOA length, and the light propa‐gation direction in the gain recovery using pump–probe techniques to qualitatively study theshortening of the gain recovery time toward achieving high bandwidth switching operations.Section 5 presents a newly introduced efficient solver in the time domain, relying on implicitschemes and multigrid concepts. Implicit schemes allow developing an adaptive step size-controlled solver for the WDM SOA response [42–44], which extends the validated TMMmodel by lifting the limitations of spatiotemporal grids and releasing adaptive sampling at theSOA output. Toward application in realistic system-level scenarios, the last section presentsexperimentally validated numerical results for two predominant SOA-based all-optical signalprocessing circuits, such as a coupled SOA XGM flip-flop arrangement and an XOR gate basedon SOA-MZI XPM configuration. The former acts as memory for sequential logic processingcircuits, while the latter forms a basic building block of combinational logic signal processingcircuits. Conclusions are addressed in the final section.

2. Established SOA numerical models

SOA numerical modeling has progressed on multiple fonts during the last decades, andvarious simulations approaches have been developed [29–36]. SOA models can be roughlydivided in two categories: (i) the material models that target optimization of the emissionproperties, before and during the development-fabrication process of an SOA device and (ii)the circuit models, which account for the carrier density, the interband, and the intrabandphenomena for simulations of light–matter interaction in realistic system-level applicationscenarios.

The material models have long studied the gain coefficients, emission characteristics, andspectral properties of the active material. Optimization parameters include the peak gain; themolecular fractions of the materials in the semiconductor compounds; the cross-sectionaldimensions, e.g., ridge width, core height, etc.; and the structural properties, e.g., number ofquantum wells (QW), quantum dots (QD), and so on. Gillner et al. [45] investigated long-wavelength semiconductor laser amplifiers by means of an experimentally validated SOAmodel and considering different structural parameters such as thickness of the active layerand amplifier length. The model that takes into account Auger recombination, thermal effects,and spontaneous emission was developed in order to optimize the spectral gain properties ofSOA lasers, such as peak gain wavelength shift and width of gain curve. Interestingly, it wasshown that there exists an optimum active layer thickness with respect to current density fora certain gain, while increased SOA length allows higher gains with reduced wavelengthvariation of the peak gain. Similarly, a comprehensive model was presented by Minch et al.[46] for the calculation of the band edge profile of both the In1–xGaxAsyP1–y and In1–x–yGaxAlyAsquantum-well systems with an arbitrary composition, as typical semiconductor compounds

Efficient and Validated Time Domain Numerical Modeling of Semiconductor Optical Amplifiers (SOAs)...http://dx.doi.org/10.5772/61801

3

found in SOAs. This model provided in-depth knowledge with mathematical curve fitsbetween the measured net modal gains for both material systems and calculations from therealistic band structure, including valence band mixing effects, allowing to extract therelationship between total current density and carrier density. More recently, research hasfocused on studying the properties of newly introduced active compound materials, such asthe GaInNAs. The analysis of the broadband gain of a GaInNAs single quantum-well (QW)SOA that takes into account the tunability of the gain and incorporates quantum dot (QD)fluctuations due to compositional fluctuations of N within the QW is presented by Xiao et al.[47]. Finally, III/V-on-SOI bonding processes have also been into design consideration recently,with a comprehensive model presented by Cheung et al. [48] providing valuable insight to thedesign requirements of the optimization process of the structural and material parameters,e.g., the width, composition, and number of quantum wells of a compressively strained In(1–x–y)Ga(x)Al(y)As quantum-well active region for emission at approximately 1550 nm.

This chapter focuses more on the 2nd category, toward presenting a holistic simulationapproach of time domain numerical modeling of circuit-level simulations. An advanced timedomain dynamic numeric model that accounts for the ultrafast gain dynamics, including theintraband phenomenon, the gain saturation, and the gain spectral profile of an SOA, has beenpresented by Toptchiyski et al. [49], where the model is employed in a system-level simulationto investigate the gain dynamics of the light-propagation direction during pump probemeasurement and application in an Sagnac interferometer switch semiconductor laseramplifier in a loop mirror (SLALOM). By exploiting the carrier density fluctuation due tointerband phenomena within the SOA, many applications have been demonstrated already.Accurate designs and numerical analysis of various SOA-based devices have resulted insuccessful demonstrations of optical flip-flops and random access memory (RAM) cells, orSOA–Mach–Zehnder interferometers (MZIs) and cutting edge optical routing devices,profiting from cross-gain modulation (XGM) or cross-phase modulation (XPM) phenomena[38]. Meanwhile, circuit-level modeling approaches have been developed for experimentalverification and accuracy.

Traditional time domain SOA models in principle treat the gain dynamics based on thelongitudinal carrier distribution along the device, relying on the free carrier rate equation forthe electron–photon interaction. To achieve this, a fine longitudinal SOA discretization isusually adopted. Dividing the SOA into cascaded elementary sectors of equally small propa‐gation length and time stepping allows applying wideband steady-state material gaincoefficients [31] across the emission spectrum, using a one-by-one space and time representa‐tion of the carrier densities and photon fluxes. This discretization derives from an explicitscheme for solving the associated differential equations, resulting in an equidistant grid withconstant time stepping. This technique is the standard approach used in circuit-level designsand has proven capable of delivering quantitative matching with experimental measurements[30–35]. Following this concept, an experimentally validated time domain SOA model relyingon the TMM analysis technique was developed by Vagionas et al. [37] with the simulationresults coming in close agreement with the characterization measurements on a commercialSOA device in terms of the emission spectrum, gain profile, and recovery time. This model


was later on extended with the plasma effect [38] to account for the effective refractive indexchange and hence the phase shift term in order to provide for a comparative study of theperformance characteristics of XGM- and XPM-based circuit topologies, indicating that theXPM phenomena are more resilient to high-speed operation.

As adopting explicit schemes for solving the system of coupled differential equations canbecome sometimes quite large even for a single SOA device, efficiency also needs to beaddressed. Especially as integration technology and circuits evolve into complex net‐works, simulating multiple SOA devices simultaneously in the time domain in complexcircuit configurations or with multi-wavelength signals travelling in various directionsfurther exacerbates the numerical requirements. Simplified analytical gain models pro‐vide only qualitative conceptual results by suggesting exploring computational efficiencyat the cost of accuracy [36]. In this regime, a few SOA models have been recently present‐ed, demonstrating techniques to speed up the simulation time at the cost of some littleaccuracy. For instance, the use of one auxiliary signal with an effect equivalent to theamplified spontaneous emissions (ASE) was presented by Vujicic et al. [32] in order toreduce the wavelength channels under study. Almost similarly, a single-state variabledepending on the available carrier densities within the SOA, termed as a “reservoir” ofexcited carriers, was presented by Mathlouthi et al. [33], resembling the reservoir of excitederbium ions in an erbium-doped fiber amplifier (EDFA). Despite the state variable, themodel still relies on an equidistant spatiotemporal grid with constant time sampling at theoutput of the SOA, even at relatively low bitrates of 1 Gb/s.

3. Accurate Transfer Matrix Method (TMM) SOA model

The TMM numerical analysis technique was initially presented by Davis et al. [39] for DFBlasers and was initially presented for the extraction of the complete transfer matrix (TM) oflarge structures with rather simple configuration. The TMM divides the SOA longitudinallyinto m small sectors of length Δz = L/m, where L is the SOA length along the propagationdirection. Considering a short sector length Δz and a short time interval Δt, the light requiresonly a small time interval Δt = ug∙Δz, where ug is the group velocity, to propagate througheach sector. During this small time interval, the structural and material parameters of eachsector, such as the carrier density and the material gain, can be considered uniform andconstant, while transverse variations are still not allowed. This allows considering a widebandsteady-state gain model.

The light propagation through a single sector can then be described by the simple TM, whichrelates the incoming and outgoing amplitudes of an elementary waveguide sector. For thecurrent analysis and without loss of generality, we assume a signal Es propagating to the rightdirection and a second signal counterpropagating toward the left side, Ec. By applying the TMof sector i at time t for both the externally injected lights and the spontaneously emitted photonsover a wide optical spectrum of 1500–1600 nm, we obtain the following equation:


5

( ) ( )

( )

( )( )

, 1, , ,

( , , ( , 1,

, , 0( , 1, 0

i

i z

i z

Es t t i i Es t i iTM

Ec t t i i Ec t i i

Es t i ieEc t i ie

g l

g l

l l

l l

l

l

D

D

é ù é ù+ D += ´ =ê ú ê ú

+ D +ê ú ê úë û ë ûé ù é ùê ú= ´ ê ú

+ê ú ê úë ûë û

(1)

where the propagation constant γ(λi) = geff/2 + j∙neffω/c describes the amplification (real part)and the phase shift (imaginary part) of the wave amplitude, after propagating through anelementary sector waveguide, as illustrated in Figure 1. Each updated wave amplitude thatemerges at the output facet of a single sector can in turn be regarded as incident to the nextsector, and the process is repeated until both lights exit the SOA. At each iteration, thewaveguide propagation constant is updated and a new operant TM is estimated. Thisoperation implies that a TM of a sector is time dependent since the matrix elements e γ(λi)Δz aresubject to the net gain coefficient geff and phase shift changes neff. Figure 1(i) depicts the lightpropagation across the SOA lengths according to the TMM, while Figure 1(ii) depicts the ithsector and the operation of the TM on the incoming and outgoing wave amplitudes.

Figure 1. (i) Longitudinal division of an SOA into m sectors of Δt and Δz time and space interval. (ii) Forward andbackward light propagation and amplification through an elementary waveguide.

For the estimation of the real part of the constant, the Connelly gain coefficient has beenemployed, which was initially presented by Connelly [31] for the estimation of a widebandsteady-state gain approximation of a bulk InP–InGaAsP homogeneous buried ridge stripeSOA, supporting large operating regimes. The gain coefficient is given by

( ) ( ) ( )322

32 22

2

4 2

ge hhm c v

e hh

Em mcg f fhm mna

n n np tn

æ öç ÷ é ù= - -ë ûç ÷+è ø�

(2)

where c is the speed of light in vacuum, na is the refractive index in the active region, τ is theradiative recombination lifetime, v is the optical frequency, me and mhh are the conduction bandelectron and valence band heavy hole effective masses, respectively, ħ is the Planck’s constantdivided by 2π, Eg is the bandgap energy and fc(v), and fv(v) is the Fermi–Dirac distributions inthe conduction and valence band, respectively. Subsequently, we incorporate a loss coefficientalong the SOA, toward a net gain coefficient given by [40]


( ) ( )net eff c s eff WG1g g a a a g aa= - G - - G - = G - (3)

where αα describes the losses in the active region, αc the losses in the cladding layer, and αs

the scattering losses m and by αWG; we denote all cumulative losses. Carrier-dependent losseshave also been presented in the literature, e.g., as given by α(N)-Κ0+ΓΚ1Ν, with K0 accountingfor the overall material and waveguide losses of Eq. (3) and the second term accounting forthe intervalence band absorption, which is current dependent. However, the constant term hasalso been shown to provide very good experimental matching. After the estimation of theupdated amplitude, amplified by the net gain coefficient through propagation by one sector,the power emerging at the output facet is given by [39]

( ) ( ) 2, ,P i t h wd E i tn= (4)

where hv is the photon energy, w is the width, and d is the thickness of the active region of theSOA, while the photon density of each stream is approximated by [39]

( ) ( ) 2,

,g

E i tS i t

u= (5)

Finally, more rigorous electron statistics and fast nonlinear phenomena of the spectral holeburning and the intraband phenomena are only taken into consideration through the use ofthe gain compression factor ε, as governed by [40]

neteff 1

ggSe

=+

(6)

Following the estimation of the photonic densities for all signals travelling through each sector,we update the carrier density Ni according to the well-known rate equation [39,40]:

( ) ( ),21 2 3 g m

1,2,3

u iii i i i

u

N J N c c N c N u g St ed =

¶= - + + - G

¶ å (7)

where J is the injection carrier density; c1, c2, and c3 are the current leakage, radiative recombi‐nation, and Auger recombination rates; Γ is the confinement factor; and Si is the photon densityat the ith sector. The index u = 1, 2, 3 refers to the ASE or the two input streams.

Having updated the carrier density, we can proceed to the estimation of the imaginary part ofthe propagation constant, which accounts for the phase shifting term of the amplitude. The


7

estimation of the effective refractive index neff in the imaginary part relies on the plasma effect,according to which the change of the refractive index Δnpl is linearly dependent on the changeof the free carrier density in the active region ΔN [40]:

2

pl 2e hheq

1 12

enm mnow e

æ öND = - +çç

è

D÷÷ø

(8)

where e is the electronic charge, ω is the angular frequency, ε0 is the permittivity of free space,neq is the effective index of the waveguide, and me and mhh are the effective mass of the electronand the heavy hole. An accurate estimation of the real and imaginary part of the propagationconstant γ(λi)=geff/2+j∙neffω/c of the ith sector allows describing the forward and backward(counter)propagating amplitudes through a simple time dependent TM given in Eq. (1).

4. Gain parameterization procedure

In this section, we report on a gain parameterization procedure and a methodology towardtailoring the TMM model and producing simulation results close to the experimental meas‐urements obtained from a commercial SOA device. The methodology is in principle genericand compatible with the other SOA devices. The target is to identify a set of material param‐eters and cross-sectional characteristics by incorporating feedback from experimental meas‐urements such as the bandgap shrinkage to fit the experimental gain spectrum in a spectrumanalyzer, the recombination rates to match the gain profile of gain-vs-incoming powermeasurements, and the gain compression factor for the recovery time at pump probe.

The procedure includes a number of steps clustered into two sections: first, the definition ofthe material and waveguide properties and, second, the feedback from the experimentalcharacterization of the SOA device under study.

The material and waveguide properties include the following:

1. To define the length of the active region of the fabricated SOA waveguide, according tofoundry.

2. To estimate of the initial bandgap energy Eg0 of the active region, assuming no externalcurrent injection

3. To define the loss coefficient of the waveguide

4. To incorporate the waveguide cross-sectional dimensions, e.g., core height and width

5. To define the external operating current I and the current density injected to the activeregion.

The cross-sectional dimensions and the length of the SOA waveguide or the active materialcan be provided by the fabrication foundry. For the definition of the material properties,


extensive literature [45–48] has been developed on III/V compounds, especially for commoncompounds such as the InGaAsP-based devices. The initial bandgap energy Eg0 when nocarriers are injected in the active region can be provided by the quadratic approximationEg0 = e(a + by + cy 2) [31], with e being the electronic charge and y the molar fraction of theArsenide in the active region, and a, b, and c being the coefficient of the quadratic approxima‐tion. Similarly, the comprehensive model presented by Cheung et al. [48] and dealing with theoptimization of the structural and material parameters of In(1–x–y)Ga(x)Al(y)As quantum-well (QW) SOAs toward bonding on an SOI platform provide an in-depth insight into bandgapenergy.

In addition to the definition of the material properties, feedback from a series of experimentalcharacterization measurements on real SOA devices can be incorporated in the model throughthe following three steps:

6. To define the bandgap shrinkage coefficient to fit the experimental gain spectrum of theSOA. The bandgap shrinkage coefficient Kg affects the blue shifting of the gain peak atthe ASE output spectrum of the SOA, which occurs when increasing the injection current.An increase in the carrier density is translated into shrinkage of the material bandgap bya term ΔEg = eKgN 1/3 [31]. In order to extract the parameter for the Kg coefficient, staticmeasurements of the ASE spectrum at the output of the SOA under different externalcurrent injection and no injected optical power are suggested.

7. To define the recombination rates for the gain curve versus the incoming optical power.This includes the c1, c2, and c3 parameters that are associated with the linear recombina‐tion coefficient at defects (current leakage), the spontaneous recombination, and the Augerrecombination rate, respectively. In practice, these three mechanisms exhibit complicateddependences that together account for the overall recombination rate given byR(N )=c1 + c2N 2 + c3N 3 [31]. This step requires static power gain measurements using acontinuous wavelength light source close to the gain peak wavelength and under varyingaverage optical power in order to estimate net gain of the SOA, and thus defining the smallsignal gain and the saturation point of the SOA.

8. To define the gain compression factor to fit the recovery time. The gain dynamicsassociated with the intraband phenomena are associated with a fast recovery timeimmediately after the injection of a short pulse, with the gain compression increased underincreasing pulse energy and bias current. Although the TMM model focuses on theinterband phenomena, the ultrafast intraband phenomena have been considered throughthe use of a power-dependent gain compression coefficient [40], which allows tuning thefast recovery time of the intraband phenomena. In this strep, typical pump probemeasurements are required with a weak Continuous Wavelength signal with averageoptical power in the small signal gain fed to an SOA along with a high-power ultrashortreturn-to-zero pulse with a full width half maximum duration of 5 ps, which sweeps allthe available carriers.


9

Considering the numerous interrelated phenomena that affect the performance of an SOA,determining the optimum parameters that match all system-level parameters, such as the gain–power curve, the spectrum, and the recovery time, implies a certain difficulty. Defining acertain methodology for parameter extraction ensures accuracy of the model for directapplicability in realistic use case scenarios and demonstrations. After developing a certain gainparameterization procedure, the use of multiobjective genetic algorithms supporting anautomated iterative procedure can also been adopted [32] toward a best-fit criterion.

4.1. Quantitative experimentally verified numerical measurements

Following an experimental characterization on a commercial SOA device [50], the previousgain parameterization procedure was adopted in order to extract the optimum simulationparameters and tailor the TMM model. Figure 2(i) illustrates the experimental setup used forthe characterization of the SOA device. Two tunable laser diodes (TLD) operating at wave‐lengths λ1 and λ2 were employed characterization of the SOA. The CW output at λ1 of thefirst TLD was amplified through an EDFA and filtered through a band-pass filter (BPF) beforebeing driven to the input of the SOA. The CW output at λ2 of the second TLD was modulatedby means of an electroabsorption modulator (EAM), whose operation was driven by anelectrical clock signal. In this way, short optical return-to-zero (RZ) pulses were producedamplified through an EDFA and filtered through a BPF at λ2 wavelength. The produced RZpulse stream was coupled with the λ1 CW and also fed to the SOA. Polarization controllers(PCs) were employed at various stages of the experimental setup to adjust the polarization ofthe optical signals as discrete off-the-shelf fiber pigtailed components were employed. Usingthe presented experimental setup, all three measurements of steps 6, 7, and 8 were conductedusing variable optical attenuators (VOAs) at the fiber-pigtailed branches before the SOA. VOAsallowed controlling the average power of the optical signals properly. In step 6, no opticalinput fed to the SOA, and both VOAs provided maximum attenuation, while the SOA outputwas recorded at an optical spectrum analyzer (OSA). By measuring a peak gain around 1560nm, at step 7, the CW signal of TLD1 was tuned at λ1 = 1555 nm, and the output of the SOAwas measured by a power meter, after filtering through a BPF.

The TMM model was in turn tailored to match the experimental measurements. Figure 2(ii)illustrates the normalized simulated optical field distribution along the SOA using the TMM,with the vertical z-axis being the optical power and the horizontal x-axis being the wavelengthrange between 1520 and 1580 nm. The plot illustrates how an input simulation signal at 1555nm wavelength of the SOA is propagated together with the generated ASE noise sector bysector, as marked in the y-horizontal axis, until it emerges at the final sector at the SOA output.Moreover, Figure 2(iii)–(v) demonstrates the simulation-experimental matching: The gainversus input power in Figure 2(iii) for a single λ1-CW signal reveals accurate matching bothin the small signal region as well as in the saturate gain regime, with less than 1 dB error foran input power between –30 and 0 dBm. Additionally, the simulated output spectrum fits wellthe experimentally recorder spectrum with less than 2 dB errors, while the matching in thetime domain of the 10%–90% gain–recovery time of the SOA reveals accurate prediction of theresponse of the SOA in the ps time scale. The simulation parameters extracted by this procedureare presented in Table 1.


Symbol Parameter description Value

m Number of longitudinal 20

Δt Temporal sector interval 1 [ps]

L SOA length 1600 [μm]

w Active layer width 1.2 [μm]

d Active layer thickness 0.1 [μm]

Γ Confinement factor 0.17

ug Group velocity 8.5·108 [m/s]

c1 Linear recombination rate 1·107 [m/s]

c2 Bimolecular recombination rate 50·10–17 [m3/s]

c3 Auger recombination rate 80·10–41 [m6/s]

awg Waveguide losses 5800 [1/m]

R Reflectivity at facets 0

I External current injection 300 [mA]

nao Reflective index in the active region 3.22

ncl Refractive index in the cladding 3.1

Eg Bandgap energy 0.7773 [eV]

Table 1. Main parameters of the SOA device used in the simulation

4.2. Qualitative results on SOA gain recovery time

To accommodate the increasing demand of data transfer and high-speed optical telecommu‐nication networks with terabit transmission capabilities and high bandwidth switchingfunctionalities, there has been a growing interest in increasing the recovery time of the SOA.High-speed nonlinear SOAs are used to perform either XGM or XPM modulation between two

Figure 2. (i) Experimental setup employed for the characterization of the SOA device for feedback to the TMM simula‐tion model. (ii) Propagation of the signal across 15 SOA sectors according to the TMM model in the 1520–1580 rangeand simulation-experimental matching for (iii) the gain versus the input optical power, (iv) the output spectrum with30 μW CW input, and (v) the recovery time of after pump–probe measurements


11

input signals, i.e., a weak CW probe signal and a high-speed data signal with a short pulsewidth, acting as pump. Increasing the external current injection or the elongating the SOAactive region have been shown to shorten the recovery time and thus support higher speedswitching operations. After developing an experimentally verified time domain model, wenumerically investigate the shortening of the recovery time during pump probe measurementsunder increasing current or increasing SOA length.

Figure 3(i) illustrates the simulated gain recovery and the normalized output power of theSOA for various external currents. The external currents were tuned from 185 to 350 mA, withthe 300 mA being the nominal current operation. The plots reveal a shortening at highercurrents, stemming from the increased current density injected to the SOA. The measured1/e gain recovery times were plotted versus the supplied current in the inset, demonstrating ashortening from 45 ps under 185 mA down to 20 ps under 350 mA.

Similarly, Figure 3(ii) illustrates the simulation results for the normalized SOA power outputand the gain recovery after pump probe measurements, while the external current injectionwas maintained constant at nominal values of 300 mA. The plots reveal a shortening of therecovery time when elongating the SOA from 0.5 to 1.7 mm. The obtained 1/e recovery timewas measured and plotted in the inset, indicating a shortening from 63 ps for a 0.5-mm-longSOA down to 20 ps for a 1.7-mm-long SOA.

Figure 3. (i) Gain recovery of the SOA for increasing external current injection ranging from 185 to 350 mA. The insetplots the measured 1/e recovery time versus the current. (ii) Gain recovery time of the SOA for different lengths of theactive region ranging between 500 and 1700 μm. The inset plots the measured 1/e recovery time versus the SOA length.

The effect of the propagation direction of the pump signal was also investigated using theTMM SOA model. Figure 4 depicts the simulated time traces of the normalized output powerand the phase shift of the probe light, obtained after pump probe measurements with the pumpcontrol pulse being fed in a co- or counterpropagating direction. The CW probe signal featuredan average power of 50 μW, which lies in the small signal regime, while the peak of the controlpump pulse was tuned, so as to induce a π phase shift at the CW signal for either propagationdirection.

The normalized traces are plotted of the XPM operation imprinted on the probe signal areillustrated in Figure 4(i) and the XGM operation in Figure 4(ii). It is obvious that in both cases,a π phase shift was introduced due to XPM operation. However, the probe signal is suppressed


more in the case of the copropagating pump pulse, reaching normalized power values downto 0.1, while for the counterpropagating pump pulse the gain suppression is less deep. Thisowes to the increase pump–probe light interaction along the copropagating direction throughthe carrier rate equation. In addition, to the recovery time in case of a counterpropagatingcontrol pulse is shorter (after the suppression of the gain a local minima), however, the trade-off is a slower response/fall time observed with a delay in the deep (minima of the outputpower) when the control pulse is injected in the counterpropagating direction. This owes againto the travelling time and the time required for the counterpropagating pump pulse to absorbthe available carriers. The findings for XGM effects and XPM effects for π-phase shift of aGaussian control pulse are shown in Figure 4 and are in full compliance with the ones foundin the literature [41].

Figure 4. (a) XGM and (b) XPM effects imprinted on a CW probe signal by a co- or counterpropagating RZ pump pulsethat induces a π phase shift.

5. Efficient multigrid SOA model with adaptive time stepping (TS)

Numerical SOA modeling has so far relied on explicit schemes for solving the associatedsystem of coupled ordinary differential equations (ODE), comprising the spatial discretizedcarrier density rate equation given in Eq. (6) combined with the material gain coefficient inEq. (2). In the literature, ODEs are classified in two categories: the stiff and nonstiff ODEs. Thelatter can be solved efficiently by explicit time stepping schemes. The former requires manysteps with explicit schemes as warranted by the smoothness of the solution, as has been thecase for the traditional SOA modeling and is schematically illustrated in Figure 2(ii) for theTMM model, where each node of the plot represents a small time step. An interesting alter‐native option would be to deploy implicit schemes, which could alleviate the problem of themany unneeded, as far as accuracy is concerned, time steps at the cost of having to determinethe Jacobian for the set of ODEs and inverting a matrix, at each time step. Implicit schemeshave shown to be more efficient for many problems [42], with multigrid methods known tobe among the most efficient solvers for many partial differential equations (PDEs) [43].


13

Vagionas and Bos [44] presented a novel multigrid solver for the dynamic response of the SOA,relying on the wideband steady-state gain coefficient of Eq. (2). Introducing multigridtechniques in SOA modeling enabled extending the accurate time domain modeling of theTMM model, allowing for the development of an efficient solution supporting implicit timediscretization schemes. Implicit schemes in turn enable accuracy—instead of stability—restricted time discretization of the signals. This implies a different discretization scheme,where sampling of the signals is optimized for certain accuracy in the solver instead of certaintime step restriction. This allows lifting the limitations of an equidistant spatiotemporal gridfor the representation of the incoming signals adopted by traditional explicit SOA models,releasing an adaptive step size-controlled solver for the dynamic SOA response with densetime sampling under a rapidly varying SOA signal output and scarce time sampling whennegligible changes are observed. Adaptive times stepping adds one more degree of freedomto computational efficiency of accurate SOA modeling, which is of crucial importance whenevaluating large input patterns for statistical signal analysis independent of the bitrate or largecircuit networks with multiple SOA-based components.

Multigrid methods employ a series of coarser grids to obtain grid independent convergencerates. Drawing from the finest grid of traditional longitudinal division of the SOA intocascaded sectors, coarse and coarser grids with less number of SOA sectors are adopted inorder to represent the spatial free carrier density distribution by less and less grid points(carrier density samples). This is schematically illustrated in Figure 5(i), where 4 grid levelshave been employed. The 4th level is the finest level, including an SOA longitudinal discreti‐zation into 16 sectors, equal to the discretization employed in the TMM model. However, byapplying multigrid concepts, the carrier densities of two neighboring sectors can be repre‐sented by a single sector in the coarser 3rd-level grid, resulting in half grid points. Equivalently,the 3rd-level can be again restricted to coarser grids, with each transition halving the numberof grid points.

Figure 5. (i) A series of four grids illustrating the finest SOA carrier density granularity of N = 16 sectors down to thecoarsest grid of N = 2 sectors, which represent the carrier density distributions along the SOA. (ii) Graphic representa‐tion of the five step multigrid V-cycle schedule. The initial finest approximation is smoothed (step 1), restricted to thecoarsest grid (step 2), solved with coarse granularity (step 3), refined to the finest grid (step 4), and finally smoothedagain (step 5).


In the following the notation, h =Δz and H=2Δz are used for the representation of the SOAsector length in the fine and coarse grid, respectively. In order to focus on the multigrid aspects,the rate equation of Eq. (6) is written in an operator form as follows:

hh h hi

i iN L N ft

¶= +

¶(9)

where the operator L, although simple in appearance, is the quite complex, as it involvessolving the forward and backward propagation equations for both the signals and noisephoton fluxes. By applying the implicit midpoint rule, we obtain the following equation:

, , 1 , , 1, , ,

2

h k h k h k h kh k h k h h ki i i i

i ik

N N N NL N L ft

- -- += - =

D(10)

where tκ = tκ−1 + Δtkand f ih ,k =0for h =Δz. The transition from a fine toward coarser grid can be

achieved through an averaging restrict operator Ih H , where a coarse grid point can beconsidered as an average of two neighboring fine points. Equivalently, the operator IH h ,Ih

H , andIHh refines the coarse grid functions to a finer grid. In order to generate an intermediate

fine point between two coarse grid points, an inverse operator has to be considered comparedto the restrict operation, which in our case is the linear approximation. In this way, a multigridcorrection V-cycle MG(N̂ h ,k , L h ,k , f h ,k )→ N̄ h ,k has been developed, described by the follow‐ing five steps:

( )( ) ( )

( )( )( )

, , , ,1

, , , , , ,

, , ,

, , , ,

, , , ,2

Step1 : smooth , , ,

Step2 : ,

Step3 : solve ,

Step4 :

Step5 : smooth

ˆ

ˆ ˆ

ˆ

, ,

ˆ

,

h k h k h k h k

h k H h k H k H k H h k H h kh h h

H k H k h k

h k h k h H k H h kH h

h k h k h k h k

N N L f

N I N f L I N I r

N f N

N N I N I N

N N L f

n

n

¬

¬ = +

®

= + -

¬

%

%

%

% % %

%

(11)

The developed multigrid V-cycle, including the above five steps, is schematically representedin Figure 5(ii), showing four grids of different granularity. Each coarse grid comprises half thegrid points compared with the higher/finer grid level, while transitions rely on the Ih

H restrictand IH h refine operators. The smooth operation exist of second-order distributive Jacobirelaxations has been considered, that is, for each, Νιh ,κ, an update is calculated as follows:

( ), , ,

, , , ,, , ,1 1

1 1/2 2

h k h k h kh k h k h k h k i i ii i i h k h k h k

i i i

dL dL dLf L NdN dN dN

d- +

æ ö= - - +ç ÷ç ÷

è ø(12)


15

The solve operation can be implemented recursively, while the solution of the coarse gridproblem is used to correct the fine grid approximation. High-frequency components owingto the interpolation of the correction cycle can be removed by the final smooth opera‐tions. The presented cycle features a grid-size-independent rate of reducing in the error bye h ,k = N̂ h ,k + N h ,keh,k = ∥N̂h,k−Nh,k∥ . When using the solution of the previous time step asinitial approximation to the solution at the current time step, k, O(log(1/h)) cycles are neededto solve the problem to the level of the truncation error. This can be reduced to O(1) byusing the coarser levels to generate an initial approximation accurate to within the level oftruncation of grid, H [43].

The proposed solver employs the rate equation (Eq. 6) and the multigrid techniques to solvethe carrier density distribution along the SOA in coarser and coarser grids, while the propa‐gation and amplification of the signal is still based on the Connelly material gain coefficientemployed in the TMM model [37]. Adopting the previous gain coefficient ensures equivalentsteady-state results, such as the optical spectrum and the net gain, and tailoring of the SOAparameters with experimental measurements. On the other hand, incorporating an implicittime discretization scheme and adaptive time sampling suggests that computational efficiencyis exploited based on the adaptive time sampling in order to benefit from long bit patterns orsmall pattern changes at the input bit-streams.

The increase or decrease of the time step is controlled by the implicit midpoint rule with adoubling scheme through the accuracy tolerance parameter ε, as depicted in Figure 6 forvarious accuracy tolerances of the multigrid solver and single block pulses at different bitrates.The simulation results depict the pulsed NRZ transmission through a single SOA with singlepulses of –25 dBm peak power centered at 1550 nm for a bitrate of 1 Gb/s in Figure 6(i) and 5Gb/s in Figure 6(ii). The accuracy tolerance ranges between 1e–2 for the plot at the top rightcorner of the graphs and 1e–5 for the bottom left plot, highlighting that a dense time samplingis adopted immediately at the rise and fall time of the pulse, where transient phenomena areobserved. On the contrary, the trailing part of a bit pulse before the bit transitions of the outputpattern, a steady state is obtained in carrier density distribution resulting in steady-state SOAgain dynamics. Thus, negligible changes are observed at the output of the SOA, revealing thanthe time step can be adapted to a larger value for enhanced computational efficiency.

The multigrid solver was also employed toward simulating the XGM operation between twosignal in pump probe measurements. A pump control bit stream is wavelength converted ona CW probe signal at 10 Gb/s bitrate, with the CW featuring an average power of –25 dBm at1550 nm and the NRZ bit stream exhibiting a peak power of– 25 dBm at 1555 nm. A customcontrol bit pattern of 001100 at 10 Gb/s was employed, as this pattern changes at odd bits (1st,3rd, and 5th bit) with fast changes in the carrier dynamics necessitating dense time sampling.On the contrary, during the even bits, when no changes are observed at the logical values ofthe input data streams, the response of the SOA is characterized by a constant steady-stateresponse and thus requires only a few time samples. This is characteristically illustrated inFigure 7(i), where the even bits necessitate only very few samples, as highlighted with a yellowmarker at the cost of a user-defined accuracy error. Adaptive time stepping can be important


for statistical performance evaluation of SOA-based circuits, especially when long patterns orcomplex structures are employed.

Figure 7. (i) Cross-gain modulation between a CW input signal of– 25 dBm peak power at 1550 nm and an NRZ streamwith a peak power of– 25 dBm at 1555 nm using the adaptive multigrid solver at 10 Gb/s. The dotted lines mark theadaptive time-stepping mechanism of the multigrid solver (ii) plot of the average number of samples per bit requiredversus the bitrate operation.

In order to evaluate the performance and computational efficiency of the solver, the operationof wavelength conversion of a control pulsed signal to a CW input signal has been simulatedat different bitrates and accuracy levels. The evaluation is based on the number of requiredsamples that describe the output of the SOA, considering an NRZ control signal following a27-1 pseudo random bit sequence (PRBS) at bitrates of 1 Gb/s, 5 Gb/s, and 10 Gb/s. The resultshave been summarized in Figure 7(ii), showing the average number of samples per bit requiredfor the overall simulated pattern versus the bitrate. Three different accuracy errors have beenconsidered, namely, 1e–3, 1e–4, and 1e–5. Especially in the case of relatively big tolerance of ε

Figure 6. Simulation results for a block pulse of –25 dBm peak power at 1550 nm wavelength input propagating alongthe SOA, illustrating the adaptive time sampling at the output of the SOA with dense time sampling after the bit tran‐sition from 0 to 1 and from 1 to 0 and scarce time sampling at the end of the bit pulse, where a constant SOA steady-state implies negligible SOA outputs. The results have been obtained with an accuracy tolerance of ε = 1e–2 (top rightcorner), ε = 1e–3 (top left), ε = 1e–4 (bottom right), and ε = 1e–5 (bottom left) for bitrates of (i) 1 Gb/s and (ii) 5 Gb/s.


17

= 1e–3 in the first column, bits are resolved by 4 and 6 samples for the bitrates of 10 Gb/s and5 Gb/s, respectively. Furthermore, for bitrates of 1 Gb/s, an average of 10 samples per bit isrequired, suggesting that the 4 samples describe efficiently the first 100 ps (for 10 Gb/soperation), additional two samples are required for the next 100 ps (for 5 Gb/s), while the restof the four samples out of ten can effectively describe the trailing 800 ps of the bit duration of1 Gb/s.

6. Circuit-level simulation and applications

Using the developed SOA numeric modeling tools, we numerically evaluated two SOA-basedcircuits predominant in all-optical signal processing: (i) an all-optical flip-flop architecture thatexploits coupled SOA waveguides, operating on XGM phenomena, and (ii) an all-optical XORgate that exploits an SOA-MZI configuration and XPM phenomena. The numerical results arecompared with the results experimentally obtained, showing very good agreement.

6.1. Coupled SOA flip-flop exploiting XGM phenomena

The proposed SR-Flip Flop relies on the bistability between a slightly and a fully saturatedregime of two travelling waveguide SOA-XGM switches [37]. Each SOA is powered by a weakCW at λ1 or λ2 wavelength, respectively. Coupled together through a 70/30 coupler as shownin Figure 8(i), they form a simplified version of an SR-Flip Flop. The coupler defines thecoupling efficiency between the two SOAs and is also used for inserting the set/reset pulses tothe SOAs and driving the SR-Flip Flop states at the outputs. The 70/30 ratio was found to bethe optimum for the trade-off between the SOA coupling efficiency and power ratio at theinput/output ports. Exploiting XGM phenomena, one SOA at a time acts as master suppressingthe other, which consequently acts as slave. Due to symmetrical setup, the role of master andslave can be interchanged and the state of the SR-Flip Flop is determined by the wavelengthof the dominating output. A logical ”1” corresponds to SOA1 being dominant and λ1 sup‐pressing the SOA2 output signal, whereas a logical ”0” is obtained when SOA2 dominates theSR-Flip Flop. Switching between the two states requires injecting proper external set or resetpulses at the dominating SOA through the corresponding branches. When a bit of logical ”1”set signal at λ1 is injected into the dominating SOA1 of logical ”1,” its gain is saturated andthe transmission of λ1 CW input signal is blocked. This allows SOA2 gain to recover, unblock‐ing the transmission of CW λ2 and switching the SR-Flip Flop state to logical ”0.” The highλ2 CW now is fed into SOA1 serving as the control signal that suppresses its gain even afterthe set pulse is extinct. As a result, SOA2 acts as master dominating over the slave SOA1, andthe SR-Flip Flop will remain in this state until a reset pulse of λ2 wavelength is fed into SOA2.This will saturate its gain and unblock the transmission of λ1 CW, switching the SR-Flip Flopback to its initial state of logical ”1.”

The flip-flop operation was verified numerically and experimentally with set and reset pulsetraces, as shown in Figure 8(ii) and (iii), respectively. The simulated output of the flip-flop at10 Gb/s for both flip-flop outputs employed the same set/reset patterns as the experimentally


employed, shown in Figure 8(iii). The experimental demonstration was performed at a low-operation speed as dictated by the 8.5-m-long fiber-pigtailed coupling stage between the twoSOAs, which seems to be the main speed-determining factor [38]. The results of Figure 8(iii)were recorded for λ1 and λ2 CW signals of 150 μW before entering the SOAs and set/resetpulses of 1.8 mW prior to reaching the 50/50 couplers and the corresponding branches of theflip-flop. The two SOAs were driven at 300 mA.

Although the experimental data results confirm successful SR-Flip Flop operation, exhibitingan average extinction ratio (ER) of 7 dB and an amplitude modulation (AM) of less than 2 dB,due to the fiber pigtailed implementation and the large fiber lengths, the operation was limitedat low operating speeds. In order to confirm that multi-Gb/s operation speed is possible, wehave applied in the numerical evaluation of the proposed SR-Flip Flop the same pattern to theexternal signals, as the ones employed experimentally, but at 10 Gb/s. CW beams of 150 μWat 1548 nm (λ1) and 1550 nm (λ2) are fed into SOA1 and SOA2, respectively, while the set/resetdata streams of 1.8 mW average power each follow the corresponding experimentallyemployed patterns. The evaluation outputs are depicted in Figure 8(ii), assuming 2-mmintermediate coupling length, and they evidently follow the exact same pattern as the exper‐imentally obtained. For SR-Flip Flop operation with noncomplementary control pulses, theyexhibit 13-dB ER and 2.4-dB AM, whereas improved performance of 15-dB ER and 1.3-dB AMis demonstrated in case of complementary set and reset pulses. Although both outputsexperience some pattern effect and a slight gain overshoot when switching state, it is clearlyshown that full switching can be achieved at 10-Gb/s operational speed.

Figure 8. (i) Coupled SOA waveguides operating as XGM switches, in a flip-flop circuit arrangement. (ii) Numericalevaluation of the flip-flop at 10 Gb/s. (iii) Experimentally obtained traces.

6.2. SOA-MZI XOR gate exploiting XPM phenomena

An SOA-MZI gate operating as an all-optical XOR gate exploiting XPM phenomena has beenboth experimentally and numerically evaluated, as shown in Figure 9. Figure 9(i) illustratesthe arrangement of the SOA-MZI XOR gate. The XOR-gate has been experimentally andnumerically evaluated with the results illustrated in Figure 9(ii) and (iii), respectively. The


19

synchronized numerical time traces demonstrated the proof of principle. Control (Ctr) signal1 at wavelength λ2 and control 2 at λ3, respectively, are fed as control signals to the SOA-MZI.The time trace obtained at λ1 wavelength at the switching output port of SOA-MZI is illus‐trated at the third row of numerical traces, where it can be seen that a “logical 1” pulse isobtained, when exclusively one of the input control bits bears a “logical 1,” while a “logical 0”pulse is obtained, when the two input control bits are equal. The experimental eye diagram ofthe XOR output is illustrated in Figure 9(ii), revealing an extinction ratio (ER) of 8 dB and anamplitude modulation (AM) of 1 dB. Equal performance has been obtained for the simulatedeye diagram of the XOR output at Figure 9(ii).

Figure 9. (i) SOA-MZI arrangement operating as an all-optical XPM-based XOR gate, (ii) experimentally obtained eyediagram at the SOA-MZI output, and (iii) synchronized simulation time traces and eye diagrams.

7. Conclusions

A holistic methodology approach on time domain numerical modeling has been demonstrat‐ed, targeting to address accuracy and efficiency. Accuracy is addressed through the develop‐ment of an experimentally validated numerical model and a gain parameterization procedure.Following the development of a validated numerical model relying on the TMM analysistechnique, qualitative results are presented so as to investigate the gain dynamics and therecovery time of the SOA during pump–probe measurements. Efficiency is sought throughthe development of a newly introduced time domain SOA modeling technique based on themultigrid concepts to introduce adaptive time stepping.

Acknowledgements

This work was supported by FP7 MC-IAPP project COMANDER (contract no. 612257).


Author details

Christos Vagionas1,2*, Jan Bos3, George T. Kanellos1,2, Nikos Pleros1,2 and Amalia Miliou1,2

*Address all correspondence to:

1 Department of Informatics, Aristotle University of Thessaloniki, Thessaloniki, Greece

2 Information Technologies Institute, Centre for Research & Technology Hellas,Thessaloniki, Greece

3 Phoenix Software, Enschede, Netherlands

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